Theorem o1p1e2

Description: 1 + 1 = 2 for ordinal numbers. (Contributed by NM, 18-Feb-2004)

	Ref	Expression
Assertion	o1p1e2	$$⊢1_{𝑜}{+}_{𝑜}{1}_{𝑜}=2_{𝑜}$$

Proof

Step	Hyp	Ref	Expression
1		1on	$$⊢1_{𝑜}\in \mathrm{On}$$
2		oa1suc	$$⊢1_{𝑜}\in \mathrm{On}\to 1_{𝑜}{+}_{𝑜}{1}_{𝑜}=\mathrm{suc}{1}_{𝑜}$$
3	1 2	ax-mp	$$⊢1_{𝑜}{+}_{𝑜}{1}_{𝑜}=\mathrm{suc}{1}_{𝑜}$$
4		df-2o	$$⊢2_{𝑜}=\mathrm{suc}{1}_{𝑜}$$
5	3 4	eqtr4i	$$⊢1_{𝑜}{+}_{𝑜}{1}_{𝑜}=2_{𝑜}$$